Quis ut Deus wrote:Or, if rivers are masculine, could they have used "fluvius" instead of "flumen?"

Yes, and likewise amnis, but both those words are much less common than flumen in classical prose.

Quis ut Deus wrote:I guess my question is, does the word in apposition with the subject have to match in gender?

No. It often will when possible, as with e.g. victrices Athenae "victorious Athens" and currus victor "the victory chariot". But there's no reason to go out of the way to make the word for "river" match gender with each river's name.

Nutkin is a squirrel and he's male (in imagined real life!) but Nutkin = Little Nut = Nucula in Latin (as I believe) which is feminine as a common noun. Is "he" now "she" in Latin for agreement purposes of adjectives and pronouns with his name as a proper noun? Or do we call him Nuculus (an invented name) or even "Nucula Bonus" (Nutkin, the Good (male squirrel))?

Nutkin is a squirrel and he's male (in imagined real life!) but Nutkin = Little Nut = Nucula in Latin (as I believe) which is feminine as a common noun. Is "he" now "she" in Latin for agreement purposes of adjectives and pronouns with his name as a proper noun? Or do we call him Nuculus (an invented name) or even "Nucula Bonus" (Nutkin, the Good (male squirrel))?

Going by the example of Scaevola "Little Lefty", I'd say to keep with Nucula but have any adjective agree with the natural gender of whichever Nutkin you happen to be talking about (Nucula bonus in this case).

Unless I'm mistaken, the origin of the name Scaevola is scaevola manus (similar to dextra often standing for dextra manus), meaning it was originally a feminine adjective. It only got its masculine designation when it became the nickname of a man, and then eventually the cognomen of his descendants. So Nucula could work on the same basis, no?